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\title{Location-based Tag Recommendation in Social Tagging Systems}


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component; formatting; style; styling;

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\section{PROBLEM STATEMENT}\label{prob}
In this section, we describe the problem we focus on in this study and introduce some notations and definitions used in this paper.

A social tagging system consists of four types of components: a set of users (taggers) $U$, a set of tags $T$ containing $N_T$ tags, a set of resources $R$ and a set of locations $L$, where $l=(l_o,l_a)\in L$. $l_o$ and $l_a$ are longitude and latitude respectively. A set of tagging information is given as $S\subseteq U\times T\times R\times L$. A quadruple $(u, t, r, l)\in S$ means that a user $u\in U$ assigns a tag $t\in T$ to a resource $r\in R$ at location $l\in L$. In this paper, we are interested in recommending a list of resources for a given query $q$, where $q=(u_q, t_q, l_q)$ indicates that $u_q$ searches $t_q$ at $l_q$. This problem can be formulated as a ranking problem and aims to get a score function $\hat{Y}: U\times T\times L\times R\rightarrow \mathbb{R}$ which could give a score $\hat{y}_{u,t,l,r}$ for every $(u,t,l,r)$ tuple. Finally, we will return a set of top $N$ resources, which is defined as:
\begin{equation}\label{eq1}
\mathcal{TN}(u_q,t_q,l_q,N):=\arg\max^{N}_{r\in R}\hat{y}_{u_q,t_q,l_q,r},
\end{equation}
where $N$ is the number of resources to be returned.

\section{LOCATION-BASED RESOURCES RECOMMENDATION}\label{sec:LDR}
In this section, we will present how to rank existing resources with location information. Our approach has two stages. At the first stage, we try to realize location division for a given query. The goal of this stage is to divide the whole world into several regions such that the given query refers to the same concept within a region while the meaning of the query is different among regions. Consequently, the location information is discrete since we look at the regions instead of longitude and latitude pairs. After location division is done, the location dimension is generated and every region corresponds to a distinct value in location dimension. Afterwards, we will present an algorithm utilizing the tensor which includes the location dimension to calculate  the score function $\hat{Y}$. We will talk about it in details later.

\subsection{Location Division}
We try to give a division of the world $L=L_1\oplus L_2\oplus\dots\oplus L_k$, where $k$ is the number of possible regions for a specific tag $t_q$\footnote{in this study, we consider all the searching words in a query as a tag} associated with the given query $q=(u_q,t_q,l_q)$. At last of this section, a function $f_L(\cdot)$: $(t,l)\rightarrow L_i (1\leq i \leq k)$ for every $t\in T$ and $l\in L$ is generated, which means for a given $t$, $f_L$ maps $l$ into one region $L_i$. Intuitively, we consider that if two locations are close, people in these two locations are more likely to have the same understanding of a tag. On the other hand, if a tag always co-occurs with the same other tags in two locations, it tends to be reasonable to assign these two locations into the same region. Thus, the following work tries to generate a dataset consisting of instances including location information and relationships of $t_q$ with other tags. For every $r,l$ pair such that $(u,t_q,r,l)\in S$ for some $u$, we have an instance $\mathbf{x}=(f_T(t_q, t_1,r,l),f_T(t_q,t_2,r,l),\dots,f_T(t_q,t_{N_T},r,l),l)$ where $f_T(t_j,t_k,r,l)$ is defined as follows:
\begin{equation}\label{eq2}
f_T(t_j,t_k,r,l)=\frac{c(t_k,r,l)-c(t_j,r,l)}{\max|c(t_k,r,l)-c(t_j,r,l)|},
\end{equation}
where $c(t,r,l)$ is the number of occurrences of $t$ in $r$ at $l$. Eq.(\ref{eq2}) measures the differences of $t_q$ with all the other tags and scales them to the interval [-1,1]. Every instance $\mathbf{x}$ consists of the relationships between $t_q$ and other tags and the location information. Thus, our dataset for location division is denoted as:
\begin{equation}\label{eq3}
\begin{split}
 \mathbf{X}=&\{\mathbf{x}=(f_T(t_q, t_1,r,l),f_T(t_q,t_2,r,l),\dots,f_T(t_q,t_{N_T},r,l),l)\\
&|\exists (u,t_q,r,l)\in S\}.
\end{split}
\end{equation}
\comment{
The advantage of this dataset generating approach is that $\mathbf{X}$ contains both positive and negative information for the following estimation, which means it includes $f_T(t_q,t_j,r,l)$ of situation where $(u,t_q,r,l)\in S\wedge(u,t_j,r,l)\notin S$ and $f_T(t_q,t_k,r,l)$ of situation where $(u,t_q,r,l)\notin S\wedge(u,t_k,r,l)\in S$.}
Note that we only consider the resources which have been annotated with $t_q$. This is because  if a tag does not appear at a location, it gives us little to no information about what it really refers to there. Thus we just ignore those resources.
Based on these observations, we want to create partitions of $\mathbf{X}$ to maximize a posteriori (MAP) estimates of parameters (we will introduce it soon). Accordingly, we have the divisions of the locations associated with $\mathbf{x}\in \mathbf{X}$. Later, we will show how to map other locations into the regions generated above.

Suppose the dataset $ \mathbf{X}$ is the mixture of k multivariate normal distributions of $d$ dimension (we will give details of how to determine k), and let $ \mathbf{Z}=(z_1,z_2,\dots,z_{N_\mathbf{X}})$ be the latent variables that determine which distributions the observations come from. $N_\mathbf{X}$ is the number of instances in $\mathbf{X}$. Thus we have:
\begin{equation}\label{eq4}
\mathbf{X}|(\mathbf{Z}=i)\sim \mathcal{N}_d(\boldsymbol{\mu}_i,\Sigma_{i}), 1\leq i \leq k.
\end{equation}
Let $\mathcal{U}=[\boldsymbol{\mu}_i]$ and $\boldsymbol{\Sigma}=[\Sigma_i]$. Also we denote that:
\begin{equation}\label{eq5}
P(\mathbf{Z}=i)=\varphi_{i}, 1\leq i \leq k,
\end{equation}
such that:
%\begin{equation}\label{eq6}
$\sum_{i=1}^k\varphi_{i}=1.$
%\end{equation}
Let $\boldsymbol{\varphi}=[\varphi_{i}]\in\mathbb{R}_{+}^{ k\times 1}$. The model parameters to be estimated is denoted as:
\begin{equation}\label{eq7}
\Theta=(\boldsymbol{\varphi},\mathcal{U},\boldsymbol{\Sigma}).
\end{equation}
The likelihood function is:
\begin{equation}\label{eq8}
\begin{split}
L(\Theta;\mathbf{X},\mathbf{Z})&=P(\mathbf{X},\mathbf{Z}|\Theta)\\
&=\prod_{j=1}^{N_\mathbf{X}}\sum_{i=1}^{k}\mathbb{I}(z_j=i)\varphi_{i} f_P(\mathbf{x}_j;\boldsymbol{\mu}_i,\Sigma_i),
\end{split}
\end{equation}
where $\mathbb{I}$ is the indicator function and $f_P$ is the probability density function of a multivariate normal.
Now we choose expectation-maximization (EM) algorithm to find MAP estimates of $\Theta$. Thus, we will get following equations to update $\boldsymbol{\varphi},\mathcal{U},\boldsymbol{\Sigma}$:
\comment{Thus, for E step, we have:
\begin{equation}\label{eq10}
\begin{split}
Q(\Theta|\Theta^{(t)})&=E(\log L(\Theta;\mathbf{X},\mathbf{Z}))\\
&=\sum_{j=1}^{N_\mathbf{X}}\sum_{i=1}^{k}P_{ji}[\log\varphi_{i}-\frac{d}{2}\log(2\pi)
 -\frac{1}{2}\log|\Sigma_i|\\
&-\frac{1}{2}(\mathbf{x}_j-\boldsymbol{\mu}_i)^\top\Sigma_i^{-1}(\mathbf{x}_j-\boldsymbol{\mu}_i)],
\end{split}
\end{equation}
where $P_{ji}$ is the conditional probability of $\mathbf{x}_j$ belonging to the $i$-th distribution, which is determined by Bayes theorem and defined as:
\begin{equation}\label{eq11}
P_{ji}=P(z_j=i|\mathbf{x}_j;\Theta^{(t)})=\frac{\varphi_{i}^{(t)}f_P(\mathbf{x}_j;\boldsymbol{\mu}_i^{(t)},\Sigma_i^{(t)})}{\sum_{i=1}^{k}\varphi_{i}^{(t)}f_P(\mathbf{x}_j;\boldsymbol{\mu}_i^{(t)},\Sigma_i^{(t)})}.
\end{equation}

In the M step, it aims to maximize $Q(\Theta|\Theta^{(t)})$ as a function of $\Theta^{(t)}$ in order to get the estimate $\Theta^{(t+1)}$. As for $\boldsymbol{\varphi}$, we have:}
\begin{equation}\label{eq12}
\begin{split}
\boldsymbol{\varphi}^{(t+1)}&=\arg\max_{\boldsymbol{\varphi}}Q(\Theta|\Theta^{(t)})\\
&=\arg\max_{\boldsymbol{\varphi}}(\sum_{j=1}^{N_\mathbf{X}}\sum_{i=1}^{k}P_{ji}^{(t)}\log \varphi_{i}^{t}),
\end{split}
\end{equation}
\comment{
So, we have:
\begin{equation}\label{eq13}
\varphi_{i}^{(t+1)}=\frac{\sum_{j=1}^{N_\mathbf{X}}P_{ji}^{(t)}}{\sum_{j=1}^{N_\mathbf{X}}\sum_{i=1}^{k}
P_{ji}^{(t)}}=\frac{\sum_{j=1}^{N_\mathbf{X}}P_{ji}^{(t)}}{N_{\mathbf{X}}}.
\end{equation}

Then we consider the estimates of $\mathcal{U}$ and $\boldsymbol{\Sigma}$.
\begin{equation}\label{eq14}
\begin{split}
(\boldsymbol{\mu}_i^{(t+1)},\Sigma_i^{(t+1)})&=\arg\max_{\boldsymbol{\mu}_i,\Sigma_i}Q(\Theta|\Theta^{(t)})\\
&=\arg\max_{\boldsymbol{\mu}_i,\Sigma_i}\sum_{j=1}^{N_{\mathbf{X}}}P_{ji}\\
&[ -\frac{1}{2}\log|\Sigma_i|
-\frac{1}{2}(\mathbf{x}_j-\boldsymbol{\mu}_i)^\top\Sigma_i^{-1}(\mathbf{x}_j-\boldsymbol{\mu}_i)].
\end{split}
\end{equation}

Further, we have equations for updating $\mathcal{U}$ and $\boldsymbol{\Sigma}$ as follows:
}
\begin{equation}\label{eq15}
\boldsymbol{\mu}_i^{(t+1)}=\frac{\sum_{j=1}^{N_\mathbf{X}}P_{ji}^{(t)}\mathbf{x}_j}{\sum_{j=1}^{N_\mathbf{X}}P_{ji}^{(t)}}
\end{equation}
and
\begin{equation}\label{eq16}
\Sigma_i^{(t+1)}=\frac{\sum_{j=1}^{N_\mathbf{X}}P_{ji}^{(t)}(\mathbf{x}_j-\boldsymbol{\mu}_i^{(t+1)})(\mathbf{x}_j-\boldsymbol{\mu}_i^{(t+1)})\top}{\sum_{j=1}^{N_\mathbf{X}}P_{ji}^{(t)}},
\end{equation}
where $P_{ji}$ is the conditional probability of $\mathbf{x}_j$ belonging to the $i$-th distribution, which is determined by Bayes theorem and defined as:
\begin{equation}\label{eq11}
P_{ji}^{(t)}=P(z_j=i|\mathbf{x}_j;\Theta^{(t)})=\frac{\varphi_{i}^{(t)}f_P(\mathbf{x}_j;\boldsymbol{\mu}_i^{(t)},\Sigma_i^{(t)})}{\sum_{i=1}^{k}\varphi_{i}^{(t)}f_P(\mathbf{x}_j;\boldsymbol{\mu}_i^{(t)},\Sigma_i^{(t)})}.
\end{equation}
When finishing EM algorithm, we already have the partitions $L_1,L_2,$ $\dots,L_k$ of $l$ which is associated with some $\mathbf{x}\in \mathbf{X}$ by using the following equation:
\begin{equation}\label{eq17}
f_L(t_q,l)=\arg\max_{L_i}\varphi_if_P(\mathbf{x};\boldsymbol{\mu}_i,\Sigma_i).
\end{equation}
For the other locations $l$, which do not have $(u,t,r,l)\in S$ for any $u$, $t$ and $r$, we determine the region for them by:
\begin{equation}\label{eq18}
f_L(t_q,l)=\arg\min_{L_i}\min_{l_j\in L_i}D(l_j,l),
\end{equation}
where $D(l_j,l)$ is the Euclidean distance.
\textbf{Algorithm~\ref{alg:1}} summarizes the above process and gives procedure of how to generate location division.

\begin{algorithm}[H]
\caption{Location Division Algorithm by Maximizing a Posteriori (\textbf{LD$_\textrm{{MAP}}$})}
 \label{alg:1}
\begin{algorithmic}[1]
\REQUIRE{$t_q$: tag used for location division; $U$: a set of users; $T$: a set of tags; $R$: a set of resources; $L$: a set of locations; $S$: a set of tagging information.
}
\ENSURE{the predictive function $f_L$.}



\STATE Generate the dataset $\mathbf{X}$ using Eqs.~(\ref{eq2}-\ref{eq3}) with $t_q$, $U$, $T$, $R$, $L$, $S$;



\STATE k=0;

\REPEAT
\STATE k=k+1;
\STATE Initialize $\Theta=(\boldsymbol{\varphi},\mathcal{U},\boldsymbol{\Sigma})$;
	\REPEAT
\STATE Update $P_{ji}$ in light of Eq.~(\ref{eq11});
\STATE Update $\boldsymbol{\varphi}$ in light of Eq.~(\ref{eq12});
\STATE Update $\mathcal{U}$ in light of Eq.~(\ref{eq15});
\STATE Update $\boldsymbol{\Sigma}$ in light of Eq.~(\ref{eq16});
           \UNTIL convergence
\UNTIL $L(\Theta;\mathbf{X},\mathbf{Z})$ does not increase

\STATE For $\forall l\in L$ and $l$ is associated with some $\mathbf{x}\in \mathbf{X}$, get $f_L(t_q,l)$ in light of Eq.~(\ref{eq17});

\STATE For $\forall l\in L$ but $l$ is not associated with any $\mathbf{x}\in \mathbf{X}$, get $f_L(t_q,l)$ in light of Eq.~(\ref{eq18});

\RETURN $f_L$ 
\end{algorithmic}
\end{algorithm}
%\vspace{-5.5mm}
\subsection{Location-sensitive Resources Ranking}
In this subsection, we aim at getting the $\hat{Y}$ at a region level. $\hat{y}_{u,t,L_i,r}$ is calculated for every $u$, $t$, $L_i$ and $r$ rather than $\hat{y}_{u,t,l,r}$. Let $f_L(t_q,l_q)=L_q$, we can rewrite our objective function in Eq.~(\ref{eq1}) as:
\begin{equation}\label{neq1}
\mathcal{TN}(u_q,t_q,L_q,N):=\arg\max^{N}_{r\in R}\hat{y}_{u_q,t_q,L_q,r},
\end{equation}
In the following paper, we will present some models to approximate and predict $\hat{Y}$. In this case, $\hat{Y}$ relies on the model parameters and we extend Bayesian ranking introduced in \cite{DBLP:conf/uai/RendleFGS09} to optimize the model parameters based on some observations. After the model parameters are updated, $\hat{Y}$ can be worked out. We first describe how we extend Bayesian ranking and apply it to determine parameters based on our generated datasets. Note that the extended Bayesian ranking is not constrained to models which we choose to predict $\hat{Y}$. Afterwards, two models are shown to approximate $\hat{Y}$ and the procedure of using extended Bayesian ranking to update the corresponding models' parameters will be given.
\subsubsection{Model Parameters Estimation}
As we assume that for a generated region, $t_q$ means somehow the same, we map $(u,t,l,r)$ to $(u,t,\l,r)$ where $f_L(t_p,l)=\l$ for every $(u,t,l,r)\in S$. Denote $S_L=\lbrace(u,t,\l,r)|\exists(u,t,l,r)\in S\wedge f_L(t_q,l)=\l\rbrace$. Intuitively, if $(u,t,L_i,r_1)\in S_L$ but $(u,t,L_i,r_2)$ $\notin S_L$, $\hat{y}_{u,t,\l,r_1}$ should be larger than $\hat{y}_{u,t,\l,r_2}$. Thus, with the observations as follows:
\begin{equation}\label{eq19}
\mathbf{Y}=\lbrace(u,t,L_i,r_1,r_2)|(u,t,L_i,r_1)\in S \wedge (u,t,L_i,r_1)\notin S \rbrace,
\end{equation}
we try to optimize the model parameters that score function $\hat{Y}$ relies on based on Bayes' theorem:
\begin{equation}\label{eq20}
p(\hat{Y}|\mathbf{Y}) \propto p(\mathbf{Y}|\hat{Y})p(\hat{Y}).
\end{equation}
Assuming the independence of the tag assignments, this results in the MAP estimator of $\hat{Y}$:
\begin{equation}\label{eq21}
\overline{\hat{Y}}=\arg\max_{\hat{Y}}\ln \prod_{(u,t,L_i,r_1,r_2)\in \mathbf{Y}}p((u,t,L_i,r_1,r_2)|\hat{Y})p(\hat{Y}).
\end{equation}
Model with model parameters $\boldsymbol{\Theta}$ which $\hat{Y}$ relies on is plugged in. We derive an estimator for $p((u,t,L_i,r_1,r_2)|\hat{Y})$ by using the score function $\hat{Y}$:
\begin{equation}\label{eq22}
p((u,t,L_i,r_1,r_2)|\hat{Y})=\sigma(\hat{y}_{u,t,\l,r_1,r_2}(\boldsymbol{\Theta})),
\end{equation}
where $\sigma$ is the logistic function $\sigma(x) =\frac{1}{1+e^{(-x)}}$ and $\hat{y}_{u,t,\l,r_1,r_2}(\boldsymbol{\Theta})$ $=\hat{y}_{u,t,\l,r_1}(\boldsymbol{\Theta})-\hat{y}_{u,t,\l,r_2}(\boldsymbol{\Theta})$. For convenience, we will write $\hat{y}_{u,t,\l,r_1,r_2}$ short for $\hat{y}_{u,t,\l,r_1,r_2}(\boldsymbol{\Theta})$. As for the prior $p(\boldsymbol{\Theta})$, we assume that the parameters are drawn from the normal distribution $\boldsymbol{\Theta} \sim \mathcal{N}(0,\sigma_{\boldsymbol{\Theta}}^{2}\mathbf{I})$. Thus, Eq.~(\ref{eq21}) can be reformulated as:
\begin{equation}\label{eq23}
\begin{split}
\overline{\boldsymbol{\Theta}}&=\arg\max_{\boldsymbol{\Theta}}\ln \prod_{(u,t,L_i,r_1,r_2)\in \mathbf{Y}}\sigma(\hat{y}_{u,t,\l,r_1,r_2})p(\boldsymbol{\Theta})\\
&=\arg\max_{\boldsymbol{\Theta}}\sum_{(u,t,\l,r_1,r_2)\in \mathbf{Y}}\ln\sigma(\hat{y}_{u,t,\l,r_1,r_2})-\lambda_{\boldsymbol{\Theta}}\lVert\boldsymbol{\Theta}\rVert^{2}_{F}.
\end{split}
\end{equation}
Since computing the full gradients is very time consuming as $\mathbf{Y}$ is very large, we choose to perform stochastic gradient descent on the randomly drawn cases. The gradient of Eq.~(\ref{eq23}) for a given case $(u,t,\l,r_1,r_2)$ with respect to a model parameter $\theta\in\boldsymbol{\Theta}$ is:
\begin{equation}\label{eq24}
\begin{split}
&\frac{\partial}{\partial\theta}(\ln\sigma(\hat{y}_{u,t,\l,r_1,r_2})-\lambda_{\boldsymbol{\Theta}}\lVert\boldsymbol{\Theta}\rVert^{2}_{F})\\ \propto& (1-\sigma(\hat{y}_{u,t,\l,r_1,r_2}))\frac{\partial}{\partial\theta}\hat{y}_{u,t,\l,r_1,r_2}-\lambda_{\theta}\theta.
\end{split}
\end{equation}
From Eq. (\ref{eq24}), we can see that only gradient of $\hat{y}_{u,t,\l,r_1,r_2}$ needs to be computed. \textbf{Algorithm~\ref{alg:2}} concludes the process of how to use Bayesian theorem and MAP to optimize model parameters.
\begin{algorithm}[H]
\caption{ Model Optimization by Using Bayesian Theorem and MAP (\textbf{BPR-OPT})}
 \label{alg:2}
\begin{algorithmic}[1]
\REQUIRE{$\mathbf{Y}$: observation dataset ;  $\boldsymbol{\Theta}$: model parameters.
}
\ENSURE{the predictive model parameters $\overline{\boldsymbol{\Theta}}$.}



\STATE Initialize $\boldsymbol{\Theta}$;


\REPEAT
\STATE Draw $(u,t,\l,r_1,r_2)$ from $\mathbf{Y}$;

\STATE $\boldsymbol{\Theta}\leftarrow \boldsymbol{\Theta}+$

$\alpha( (1-\sigma(\hat{y}_{u,t,\l,r_1,r_2}))\frac{\partial}{\partial\boldsymbol{\Theta}}\hat{y}_{u,t,\l,r_1,r_2}-\lambda_{\boldsymbol{\Theta}}\boldsymbol{\Theta})$;
\UNTIL convergence



\RETURN $\overline{\boldsymbol{\Theta}}$ 
\end{algorithmic}
\end{algorithm}
\subsubsection{Models for Predicting $\hat{Y}$}
Factorization models are witnessed as a very popular and successful model class for recommendation and resources searching systems \cite{DBLP:conf/nips/SalakhutdinovM07,DBLP:conf/kdd/RendleMNS09}. One of the prominent models is Tucker decomposition which has been widely used in recommendation systems. Thus, instead of simplely using Euclidean distance to compute the similarity of tag pairs and matching resources with generated concepts in section~\ref{moti}, we choose to employ Factorization models. In the following, we will describe and extend existing 3D Tucker decomposition and one of its variation models to 4D versions to approximate $\hat{Y}$ and give details of how these models' parameters can be learned with \textbf{BPR-OPT}. All these models try to present $\hat{Y}$ which can be seen as the 4-dimensional tensor.
\subsubsection{Four-dimensional Tucker Decomposition}
Tucker Decomposition (TD) factorizes a high-order cube into a core tensor and one factor matrix for each dimension:
\begin{equation}\label{neq2}
\hat{Y}^{TD}=\hat{C}\times\hat{U}\times\hat{T}\times\hat{L}\times\hat{R},
\end{equation}
which has the following model parameters:
\begin{equation}\label{neq3}
\begin{split}
\hat{C}\in \mathbb{R}^{k_u\times k_t\times k_l \times k_r}, \hat{U}\in \mathbb{R}^{|U|\times k_u},\\
\hat{T}\in \mathbb{R}^{|T|\times k_t},\hat{L}\in \mathbb{R}^{k\times k_l},\hat{R}\in \mathbb{R}^{|R|\times k_r}.
\end{split}
\end{equation}
In order to derive the gradient of model parameters in a more clear way, we give an alternative equivalent representation of Eq. (\ref{neq2}):
\begin{equation}\label{eq25}
\hat{y}_{u,t,\l,r}^{\text{TD}}=\sum_{\tilde{u}}\sum_{\tilde{t}}\sum_{\tilde{\l}}\sum_{\tilde{r}}\hat{c}_{\tilde{u},\tilde{t},\tilde{\l},\tilde{r}}\hat{u}_{u,\tilde{u}}\hat{t}_{t,\tilde{t}}\hat{\l}_{\l,\tilde{\l}}\hat{r}_{r,\tilde{r}}.
\end{equation}
For learning the model parameters using \textbf{BPR-OPT}, the gradients $\frac{\partial}{\partial\theta}\hat{y}_{u,t,\l,r_1,r_2}$ used in Eq.~(\ref{eq24}) are:
\begin{equation}\label{eq26}
\begin{split}
\frac{\partial\hat{y}_{u,t,\l,r}^{\text{TD}}}{\partial\hat{c}_{\tilde{u},\tilde{t},\tilde{\l},\tilde{r}}}&=\hat{u}_{u,\tilde{u}}\hat{t}_{t,\tilde{t}}\hat{\l}_{\l,\tilde{\l}}\hat{r}_{r,\tilde{r}}\\
\frac{\partial\hat{y}_{u,t,\l,r}^{\text{TD}}}{\partial\hat{u}_{u,\tilde{u}}}&=\sum_{\tilde{t}}\sum_{\tilde{\l}}\sum_{\tilde{r}}\hat{c}_{\tilde{u},\tilde{t},\tilde{\l},\tilde{r}}\hat{t}_{t,\tilde{t}}\hat{\l}_{\l,\tilde{\l}}\hat{r}_{r,\tilde{r}}\\
\frac{\partial\hat{y}_{u,t,\l,r}^{\text{TD}}}{\partial\hat{t}_{t,\tilde{t}}}&=\sum_{\tilde{u}}\sum_{\tilde{\l}}\sum_{\tilde{r}}\hat{c}_{\tilde{u},\tilde{t},\tilde{\l},\tilde{r}}\hat{u}_{u,\tilde{u}}\hat{\l}_{\l,\tilde{\l}}\hat{r}_{r,\tilde{r}}\\
\frac{\partial\hat{y}_{u,t,\l,r}^{\text{TD}}}{\partial\hat{\l}_{\l,\tilde{\l}}}&=\sum_{\tilde{u}}\sum_{\tilde{t}}\sum_{\tilde{r}}\hat{c}_{\tilde{u},\tilde{t},\tilde{\l},\tilde{r}}\hat{u}_{u,\tilde{u}}\hat{t}_{t,\tilde{t}}\hat{r}_{r,\tilde{r}}\\
\frac{\partial\hat{y}_{u,t,\l,r}^{\text{TD}}}{\partial\hat{r}_{r,\tilde{r}}}&=\sum_{\tilde{u}}\sum_{\tilde{t}}\sum_{\tilde{\l}}\hat{c}_{\tilde{u},\tilde{t},\tilde{\l},\tilde{r}}\hat{u}_{u,\tilde{u}}\hat{t}_{t,\tilde{t}}\hat{\l}_{\l,\tilde{\l}}.
\end{split}
\end{equation}
Let $k_{max}=min(k_u,k_t,k,k_r)$. Thus, the runtime complexity of TD in this case is $O(k_{max}^4)$. 
\comment{
\begin{algorithm}[H]
\caption{ \textbf{BPR-OPT} for TD}
 \label{alg:3}
\begin{algorithmic}[1]
\REQUIRE{$\mathbf{Y}$: observation dataset ;  $\hat{c}_{\tilde{u},\tilde{t},\tilde{\l},\tilde{r}}$, $\hat{u}_{u,\tilde{u}}$, $\hat{t}_{t,\tilde{t}}$, $\hat{\l}_{\l,\tilde{\l}}$, $\hat{r}_{r,\tilde{r}}$: model parameters.
}
\ENSURE{the predictive model parameters $\hat{c}_{\tilde{u},\tilde{t},\tilde{\l},\tilde{r}}$, $\hat{u}_{u,\tilde{u}}$, $\hat{t}_{t,\tilde{t}}$, $\hat{\l}_{\l,\tilde{\l}}$, $\hat{r}_{r,\tilde{r}}$.}



\STATE Initialize  $\hat{c}_{\tilde{u},\tilde{t},\tilde{\l},\tilde{r}}$, $\hat{u}_{u,\tilde{u}}$, $\hat{t}_{t,\tilde{t}}$, $\hat{\l}_{\l,\tilde{\l}}$, $\hat{r}_{r,\tilde{r}}$;


\REPEAT
\STATE Draw $(u,t,\l,r_1,r_2)$ from $\mathbf{Y}$;

\STATE $\hat{y}_{u,t,\l,r_1,r_2}=\hat{y}_{u,t,\l,r_1}-\hat{y}_{u,t,\l,r_2}$;

\STATE $\gamma=1-\sigma(\hat{y}_{u,t,\l,r_1,r_2})$;

\FOR{$i_1=1 \to k_u$}
	\FOR{$i_2=1 \to k_t$}
		\FOR{$i_3=1 \to k$}
			\FOR{$i_4=1 \to k_r$}
				\STATE $\hat{c}_{\tilde{u},\tilde{t},\tilde{\l},\tilde{r}}=\hat{c}_{\tilde{u},\tilde{t},\tilde{\l},\tilde{r}}+\alpha(\gamma(\hat{u}_{u,\tilde{u}}\hat{t}_{t,\tilde{t}}\hat{\l}_{\l,\tilde{\l}}\hat{r}_{r_1,\tilde{r}}-\hat{u}_{u,\tilde{u}}\hat{t}_{t,\tilde{t}}\hat{\l}_{\l,\tilde{\l}}\hat{r}_{r_2,\tilde{r}})-\lambda\cdot \hat{c}_{\tilde{u},\tilde{t},\tilde{\l},\tilde{r}} )$;
			\ENDFOR
		\ENDFOR
	\ENDFOR
\ENDFOR
\UNTIL convergence



\RETURN $\overline{\boldsymbol{\Theta}}$ 
\end{algorithmic}
\end{algorithm}
}

\subsubsection{Four-dimensional Pairwise Interaction Tensor Factorization}

A serious drawback of TD is the high time complexity. This results in a variation Pairwise Interaction Tensor Factorization (PITF) proposed in \cite{DBLP:conf/wsdm/RendleS10}. PITF just considers the two-way interactions between $r$ and other dimensions. As we extend it to 4D version, the following equation can be got:
\begin{equation}\label{eq27}
\hat{y}_{u,t,\l,r}^{\text{PITF}}=\sum_{p}\hat{u}_{u,p}\cdot\hat{r}_{r,p}^{U}+\sum_{p}\hat{t}_{t,p}\cdot\hat{r}_{r,p}^{T}
+\sum_{p}\hat{\l}_{\l,p}\cdot\hat{r}_{r,p}^{L}
\end{equation}
with model parameters:
\begin{equation*}
\begin{split}
\hat{U}\in \mathbb{R}^{|U|\times p}, \hat{T}\in \mathbb{R}^{|T|\times p}, \hat{L}\in \mathbb{R}^{k\times p},\\
\hat{R}^{U}\in\mathbb{R}^{|R|\times p}, \hat{R}^{T}\in\mathbb{R}^{|R|\times p}, \hat{R}^{L}\in\mathbb{R}^{|R|\times p}.
\end{split}
\end{equation*}
From Eq.~(\ref{eq27}), we can derive the gradients of PITF as follows:
\begin{equation}\label{eq28}
\begin{split}
\frac{\partial \hat{y}_{u,t,\l,r}^{\text{PITF}}}{\partial \hat{u}_{u,p}}=\hat{r}_{r,p}^{U}, 
\frac{\partial \hat{y}_{u,t,\l,r}^{\text{PITF}}}{\partial \hat{t}_{t,p}}=\hat{r}_{r,p}^{T}, 
\frac{\partial \hat{y}_{u,t,\l,r}^{\text{PITF}}}{\partial \hat{\l}_{\l,p}}=\hat{r}_{r,p}^{L}, \\
\frac{\partial \hat{y}_{u,t,\l,r}^{\text{PITF}}}{\partial \hat{r}_{r,p}^{U}}=\hat{u}_{u,p}, 
\frac{\partial \hat{y}_{u,t,\l,r}^{\text{PITF}}}{\partial \hat{r}_{r,p}^{T}}=\hat{t}_{t,p}, 
\frac{\partial \hat{y}_{u,t,\l,r}^{\text{PITF}}}{\partial \hat{r}_{r,p}^{L}}=\hat{\l}_{\l,p}.
\end{split}
\end{equation}
Apparently, the runtime of \textbf{BPR-OPT} using PITF model is linear in $O(p)$. After we get the gradients of parameters, we can utilize them for updating these parameters according to Eq.~(\ref{eq24}) when running \textbf{BPR-OPT}. After \textbf{BPR-OPT} is done, $\hat{Y}$ could be computed based on the parameters in light of Eq.~(\ref{eq25}) or Eq.~(\ref{eq27}). Finally, $\mathcal{TN}(u_q,t_q,L_q,N)$ can be worked out using $\hat{Y}$.




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\section{Conclusion}
The conclusion goes here. this is more of the conclusion

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Acknowledgements to be added to camera ready version upon acceptance.


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\bibitem{IEEEhowto:kopka}
H.~Kopka and P.~W. Daly, \emph{A Guide to \LaTeX}, 3rd~ed.\hskip 1em plus
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